Exercises on linear transformations and their matrices
Problem 30.1: Consider the transformation T that doubles the distance
between each point and the origin without changing the direction from
the origin to the points. In polar coordinates this is described by
T (r, θ ) = (2r, θ ).
a) Yes or no: is T a linear transformation?
b) Describe T using Cartesian (xy) coordinates. Check your work by con­
firming that the transformation doubles the lengths of vectors.
c) If your answer to (a) was ”yes”, find the matrix of T. If your answer to
(a) was ”no”, explain why the T isn’t linear.
Solution:
a) Yes. In terms of vectors, T (v) = 2v, so T (v1 + v2 ) = 2v1 + 2v2 =
T (v1 ) + T (v2 ) and T (cv) = 2cv = cT (v).
��
��
� x � �
� =
x2 + y2 and can calculate
b) T ( x, y) = (2x, 2y). We know
��
�
y
� ��
��� ��
��
�
�
�
�
� � 2x
� �
�
�
�
x
� T
� =
�
� =
4( x2 + y2 ) = 2 � x
� .
�
� � 2y �
�
y
�
y
This confirms that T doubles the lengths of vectors.
�
c) We can use our answer to (b) to find that the matrix of T is
2 0
0 2
�
.
Problem 30.2: Describe a transformation which leaves the zero vector
fixed but which is not a linear transformation.
Solution: If we limit ourselves to “simple” transformations, this is not an
easy task!
1
One way to solve this is to find��
a transformation
�� �
�that acts differently
x
x
on different parts of the plane. If T
=
then
y
|y|
��
T
1
1
�
�
+
1
−1
��
�
�
=
2
0
��
�
2
2
is not equal to
��
T
1
1
��
��
+T
1
−1
=
�
.
Another approach
�� �� is to
� use�a nonlinear
� �function
�� to� define� the transfor­
x
x
mation: if T
=
y
y2
�
�� �� �
x
cx
to cT
=
.
2
y
then T c
cy
2
x
y
=
cx
c2 y2
is not equal
MIT OpenCourseWare
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18.06SC Linear Algebra
Spring 2011
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